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Vojtěch Jarník IMC
2015 VJIMC
4
VJIMC 2015 Category II, Problem 4
VJIMC 2015 Category II, Problem 4
Source: VJIMC2015
August 10, 2015
integration
college contests
real analysis
Problem Statement
Problem 4 Find all continuously differentiable functions
f
:
R
→
R
f : \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
, such that for every
a
≥
0
a \geq 0
a
≥
0
the following relation holds:
∭
D
(
a
)
x
f
(
a
y
x
2
+
y
2
)
d
x
d
y
d
z
=
π
a
3
8
(
f
(
a
)
+
sin
a
−
1
)
,
\iiint \limits_{D(a)} xf \left( \frac{ay}{\sqrt{x^2+y^2}} \right) \ dx \ dy\ dz = \frac{\pi a^3}{8} (f(a) + \sin a -1)\ ,
D
(
a
)
∭
x
f
(
x
2
+
y
2
a
y
)
d
x
d
y
d
z
=
8
π
a
3
(
f
(
a
)
+
sin
a
−
1
)
,
where
D
(
a
)
=
{
(
x
,
y
,
z
)
:
x
2
+
y
2
+
z
2
≤
a
2
,
∣
y
∣
≤
x
3
}
.
D(a) = \left\{ (x,y,z)\ :\ x^2+y^2+z^2 \leq a^2\ , \ |y|\leq \frac{x}{\sqrt{3}} \right\}\ .
D
(
a
)
=
{
(
x
,
y
,
z
)
:
x
2
+
y
2
+
z
2
≤
a
2
,
∣
y
∣
≤
3
x
}
.
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